- #1
maCrobo
- 51
- 1
Homework Statement
First exercise:
Compute the surface area of that portion of the sphere x2+y2+z2=a2 lying within the cylinder x2+y2=ay, where a>0.
Second one:
Compute the area of that portion of the surface z2=2xy which lies above the first quadrant of the xy-plane and is cut off by the planes x=2 and y=1.
Homework Equations
[itex]\int_{}^{}{\int_{}^{}{\left| \left| \frac{\partial r}{\partial x}\times \; \frac{\partial r}{\partial y} \right| \right|\partial x\partial y}}[/itex]
The Attempt at a Solution
First (exercise) parametrization: [itex]r\left( x,y \right)=x\; i+y\; j+\sqrt{a^{2}-x^{2}-y^{2}} \; k[/itex]
By using the "relevant equation" I wrote before, I get the following integral: [itex]\int_{}^{}{\int_{}^{}{\frac{a}{\sqrt{a^{2}-x^{2}-y^{2}}}\partial x\partial y}}[/itex]
Second (exercise) parametrization: [itex]r\left( x,y \right)=x\; i+y\; j+\sqrt{2xy} \; k[/itex]
By using again the "relevant equation" I wrote before, I get the following integral: [itex]\int_{}^{}{\int_{}^{}{\frac{x+y}{\sqrt{2xy}} \partial x \partial y}}[/itex]
The boundaries of the integrals can be easily got from the text of the exercise, but I won't write them not to influence your reasoning. I think there are my mistakes.
Anyway, my results are wrong, so I please you to try and tell me your results so to understand where I got lost.
Thanks in advance! :D